© 4 July 2001 – B.W. Bequette

Module 2

Introduction to SIMULINK

Although the standard MATLAB package is useful for linear systems analysis, SIMULINK is farmore useful for control system simulation. SIMULINK enables the rapid construction andsimulation of control block diagrams. The goal of the tutorial is to introduce the use ofSIMULINK for control system simulation. The version available at the time of writing of thistextbook is SIMULINK 4, part of Release 12 (including MATLAB 6) from MATHWORKS. Theversion that you are using can be obtained by entering ver in the MATLAB Command Window.

The easiest way to learn how to use SIMULINK is to implement each step of the tutorial, ratherthan simply reading it. The basic steps to using SIMULINK are independent of the platform (PC,MAC, UNIX, LINUX…).

M2.1 Background

M2.2 Open-loop Simulations

M2.3 Closed-loop Simulations

M2.4 Developing Alternative Controller Icons

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M2.1 Background

The first step is to startup MATLAB on the machine you are using. In the Launch Pad window ofthe MATLAB desktop, select SIMULINK and then the SIMULINK Library Browser. A number ofoptions are listed, as shown in Figure M2.1. Notice that Continuous has been highlighted; thiswill provide a list of continuous function blocks available. Selecting Continuous will provide thelist of blocks shown in Figure M2.2. The ones that we often use are Transfer Fcn and State-Space.

Figure M2.1 SIMULINK Library Browser.

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Figure M2.2 SIMULINK Continuous Blocks.

Selecting the Sources icon yields the library shown in Figure M2.3. The most commonly usedsources are Clock (which is used to generate a time vector), and Step (which generates a stepinput).

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Figure M2.3 SIMULINK Sources.

The Sinks icon from Figure M2.1 can be selected to reveal the set of sinks icons shown inFigure M2.4. The one that we use most often is the To Workspace icon. A variable passed to this

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icon is written to a vector in the MATLAB workspace. The default data method shoulb bechanged from “structure” to “matrix” in order to save data in an appropriate form for plotting.

Figure M2.4 SIMULINK Sinks.

M2.2 Open-Loop Simulation

You now have enough information to generate an open-loop simulation. The Clock, simout,step and Transfer function blocks can be dragged to a model (.mdl) workspace, as shown inFigure M2.5a. Renaming the blocks and variables, and connecting the blocks, results in themodel shown in Figure M2.5b. The transfer function studied is the Van de Vusse reactor(Module 5).

The s-polynomials in the process transfer function were entered by double-clicking on thetransfer function icon and entering the coefficients for the numerator and denominator

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polynomials. Notice also that the default step (used for the step input change) is to step from avalue of 0 to a value of 1 at t = 1. These default values can be changed by double-clicking thestep icon. The simulation parameters can be changed by going to the Simulation “pull-down”menu and modifying the stop time (default = 10) or the integration solver method (default= ode45).

1

s+1

Transfer Fcn

simout

To Workspace2

simout

To Workspace1

simout

To Workspace

Step

Clock

a. Placement of function blocks

t

time

stepinput

y

measured output

u

manipulated input

-1.117s+3.1472

s +4.6429s+5.38212

Transfer Fcn

Clock

b. Renaming and connection of blocks

Figure M2.5 Development of an Open-loop Simulation.

The reader should generate simulations and observe the “inverse response” behavior of theoutput with respect to a step input change. Use the subplot command to place the processoutput (y) on the top plot, and the manipulated input (u) on the bottom plot. Perform this now. Ifdesired, change the default simulation stop time by selecting the parameters “pull down” menu.

M2.3 Feedback Control Simulation

The Math icon from Figure M2.2 can be selected, resulting in the functions shown in FigureM2.6. Additional icons can be found by selecting the Simulink Extras icon in Figure M2.1.Selecting the Additional Linear icon from this group yields the set of icons in Figure M2.7.The most useful icon here is the PID Controller. Any icon can be “dragged” into the untitledmodel workspace. In Figure M2.8 we show the preliminary stage of the construction of a controlblock diagram, where icons have been dragged from their respective libraries into the untitledmodel workspace.

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Figure M2.6 SIMULINK Math.

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Figure M2.7 SIMULINK Additional Linear.

The labels (names below each icon) can easily be changed. The default parameters for each iconare changed by double clicking the icon and entering new parameter values. Also, connectionscan be made between the outputs of one icon and inputs of another. Figure M2.8b shows how theicons from Figure M2.8a have been changed and linked together to form a feedback controlblock diagram.

It should be noted that the form of the PID control law used by the SIMULINK PID Controllericon is not the typical form that we use as process control engineers. The form can be found bydouble-clicking the icon to reveal the following controller transfer function representation

gc s( ) = P+I

s+ Ds

while we normally deal with the following PID structure

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gc s( ) = kc 1+1

Is+ Ds

so the following values must be entered in the SIMULINK PID Controller

P = kc I = kc I D = kc D

1

s+1

Transfer Fcn

simout

To Workspace3

simout

To Workspace2

simout

To Workspace1

simout

To Workspace

SumStep

Clock

a. Preliminary Block Diagram

t

time

step setpoint

r

setpoint

y

output

u

manipulated

Sum

-1.117s+3.1472

s +4.6429s+5.38212

Process

PID

PID Controller

Clock

b. Completed Block Diagram, with Name and Parameter Changes

Figure M2.8. Block Diagram for Feedback Control of the Van de Vusse CSTR

The s-polynomials in the process transfer function were entered by double-clicking on thetransfer function and entering the coefficients for the numerator and denominator polynomials.Notice also that the default step (used for the step setpoint change) is to step from a value of 0 to

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a value of 1 at t = 1. These default values can be changed by double-clicking the step icon. Thesimulation parameters can be changed by going to the Simulation “pull-down” menu andmodifying the stop time (default = 10) or the integration solver method (default = ode45).

The controller tuning parameters of kc = 1.89 and τI = 1.23 are used by entering P = 1.89 and I =1.89/1.23 in the default PID Controller block. The following plot commands were used togenerate Figure M2.9.

» subplot(2,1,1),plot(t,r,'--',t,y)» xlabel('t (min)')» ylabel('y (mol/l)')» subplot(2,1,2),plot(t,u)» xlabel('t (min)')» ylabel('u (min^-1)')

Figure M2.9. Measured Output and Manipuated Input Responses to a Unit Step SetpointChange.

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This brief tutorial has gotten you started in the world of SIMULINK-based control blockdiagram simulation. You may now easily compare the effect of different tuning parameters, ordifferent formulations of a PID controller (“ideal” vs. “real”, for example).

Let’s say you generated responses for a set of tuning parameters that we call case 1 forconvenience. You could generate time, input and output vectors for this case by setting:

» t1 = t;» y1 = y;» u1 = u;

after running the case 1 values. You could then enter case 2 values, run another simulation, andcreate new vectors

» t2 = t;» y2 = y;» u2 = u;

then, compare the case 1 and case 2 results by:

» subplot(2,1,1),plot(t1,y1,t2,y2,'--')» xlabel('t (min)')» ylabel('y (mol/l)')» subplot(2,1,2),plot(t1,u1,t2,u2,'--')» xlabel('t (min)')» ylabel('u (min^-1)')

which automatically plots case 1 as a solid line and case 2 as a dashed line.

Similarly, you could modify the controller type by placing a transfer function block for thecontroller and using a “real PID” transfer function (this only differs when there is derivativeaction).

Other Commonly Used Icons

Often you will want to simulate the behavior of systems that have time delays. The TransportDelay icon can be selected from the Continuous library shown in Figure M2.2. The transportdelay icon is shown in Figure M2.10. Our experience is that simulations can become somewhat“flaky” if 0 is entered for a transport delay. We recommend that you remove the transport delayblock for simulations where no time-delay is involved.

TransportDelay

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Figure M2.10. Transport Delay Icon

Manipulated variables are often constrained to between minimum (0 flow, for example) andmaximum (fully open valve) values. A saturation icon from the Nonlinear library can be usedto simulate this behavior. The saturation icon is shown in Figure M2.11.

Saturation

Figure M2.11. Saturation Element

Actuators (valves) and sensors (measurement devices) often have additional dynamic lags thatcan be simulated by transfer functions. These can be placed on the block diagram in the samefashion that a transfer function was used to represent the process earlier.

It should be noted that icons can be “flipped” or “rotated” by selecting the icon and going to theformat “pull-down” menu and selecting Flip Block or Rotate Block. The block diagram ofFigure M2.8 has been extended to include the saturation element and transport delay, as shownin Figure M2.12.

t

time

step setpoint

r

setpoint

y

output

u

manipulated

TransportDelay

Sum Saturation

-1.117s+3.1472

s +4.6429s+5.38212

Process

PID

PID Controller

Clock

Figure M2.12. Block Diagram with Saturation and Time-Delay Elements

The default data method for the “to workspace” blocks (r,t,u,y in Figure M2.12) must bechanged from “structure” to “matrix” in order to save data in an appropriate form for plotting.

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M2.4 Developing Alternative Controller Icons

It was noted earlier that the default SIMULINK PID controller block uses a different form thanthat used by most process engineers. It is easy to generate new PID controller blocks as shownbelow. The default PID controller icon is shown in Figure M2.13a. This is “unmasked” to yieldthe diagram shown in Figure M2.13b. Again, this has the form

gc s( ) = P+I

s+ Ds

while we normally prefer the following PID structure

gc s( ) = kc 1+1

Is+ Ds

Of course, the two algorithms are related by

P = kc I = kc I D = kc D

but it would be much less confusing to work with our standard form.

PID

PID Controller

a. Block

1

Out_1Sum

P

Proportional

I

s

Integral

du/dt

Derivative

D

D

1

In_1

b. Unmasked

Figure M2.13 Default Ideal PID Controller

We have generated a new ideal analog PID, as shown in Figure M2.14a. Notice that there aretwo inputs to the controller, the setpoint and the measured output, rather than just the error signal

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that is the input to the default SIMULINK PID controller. Our new implementation is unmaskedin Figure M2.14b to reveal that

gc s( ) = kc 1+1

Is+ Ds

is the algorithm used.

r

yuPID

Ideal analog PID

a. Block

1

u

taud

taud

kc

kcSum1Sum

1

s

Integral

du/dt

Derivative

1/taui

1/taui

2

y

1

r

b. Unmasked BlockFigure M2.14 Preferred Implementation of Ideal PID Controller

Summary

SIMULINK is a very powerful block diagram simulation language. Simple simulations,including the majority of those used as examples in this textbook, can be set-up rapidly (in amatter of minutes). The goal of this module was to provide enough of an introduction to get youstarted on the development of open- and closed-loop simulations. With experience, thedevelopment of these simulations will become second-nature. It is recommended that youperform the simulations shown in this module, as well as the practice exercises, to rapidlyacquire these simulation skills.

Practice Exercises

1. Compare step responses of the transfer function and state space models for the Van de VusseReactor.

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a. Transfer function: Perform the open-loop simulation for a step input change from 0 to 1 at t =1 in Figure M2.5.

b. State space: Replace the transfer function block of Figure M2.5 with a state space block. Enterthe following matrices in the MATLAB command window.

A =−2.4048 0

0.83333 −2.2381

B =

7 0.5714

−1.117 0

C = 0 1[ ] D = 0 0[ ]

Show that the resulting step responses of the transfer function and state space models areidentical.

2. Consider the closed-loop SIMULINK diagram for the Van de Vusse reactor (Figure M2.8).Replace the default PID controller with a “Real PID” controller, as shown below in both themasked and unmasked versions. How do your results compare with ones obtained using thedefault PID controller?

r

yuReal PID

Real analog PID

1

u

kc

kc

taud.s+1

alpha*taud.s+1

derivativeSum

taui.s+1

taui.s

Integral2

y

1

r